94. A nonlinear observable state with a curved moment domain
On this page · Permalink · Source line 3764
For \(u\in[-1,1]\), put
The normalized states are moments of a scalar signal:
Their cone has the finite curved description
Assume that the declared probability measure \(\mu\) on \([-1,1]\) has moments zero at orders one and three, \(1/3\) at order two and \(1/5\) at order four. Define a two-parameter degree-four operator density
Here \(a,b\) are parameters, not action labels. This example is autonomous. Its dependence on the observed signal is nonlinear because the state includes its square.
Proposition 83: Structured curved-domain certificate
The set \(K\) is the convex hull of all \(v(u)\). If \(|a|+|b|\leq1-\rho\) for \(0\leq\rho\leq1\), then \(B_{a,b}(u)\) preserves \(C\) and has exact integrated mass. On a normalized state, its density is
Every positive-density posterior is exactly \(v(u)\). In particular, starting from \(v(s)\) produces the observable Markov law
Its next conditional moments satisfy
Read proof · Recorded Lean scope · Visual guide: Uncertainty on a curved domain · Visual guide: A confidence set can meet an exact certificate · Permalink · Source line 3798
Proof
Every distribution on \([-1,1]\) has \(x^2\leq y\leq1\). Conversely, fix \(|x|<1\). A point mass at \(x\) gives \((x,x^2)\), while the endpoint mixture with mean \(x\) gives \((x,1)\). Their convex combinations give every intermediate \(y\). The cases \(x=\pm1\) are immediate. Homogenizing yields the matrix and scalar inequalities, which also force the only mass-zero cone element to be zero.
On \(K\), \(|x|\leq1\), \(0\leq y\leq1\), \(|u|\leq1\) and \(|L_2(u)|\leq1\). Thus the density is at least \(1-|a|-|b|\). Each operator output is that nonnegative scalar times \(v(u)\), so lies in \(C\). The first and second reference moments imply \(\lambda\int B_{a,b}(u)\,\mu(du)=\lambda\). Normalization cancels the scalar density, yielding \(v(u)\). Symmetry through order three gives \(\int uL_2(u)d\mu=0\). The second and fourth moments give \(\int L_2(u)^2d\mu=1/5\). Integrating \(u\) and \(L_2\) against the density proves the last two formulas. \(\square\)
The posterior ray lies on a curved boundary. Arbitrary perturbations of its output coordinates can destroy validity, as in Proposition 72. Parameter uncertainty inside the declared family behaves differently: it changes the likelihood but not the posterior ray. A certified rectangle in \((a,b)\) can therefore be robust even though a free matrix-entry ball has no positive validity radius.
This is an example of uncertainty that preserves the model's algebraic structure. It is not a claim that every real system obeys the two-parameter family. The feature dictionary \(1,u,u^2\), its geometry, the reference measure and the parameter model are declared inputs to learning.